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Restricting Modular Spin Representations of Symmetric and Alternating Groups to Young‐Type Subgroups
Author(s) -
Phillips Aaron M.
Publication year - 2004
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611504014893
Subject(s) - mathematics , modular representation theory , symmetric group , combinatorics , algebraically closed field , simple group , type (biology) , representation theory , classification of finite simple groups , irreducible representation , simple (philosophy) , affine transformation , pure mathematics , group theory , group of lie type , philosophy , epistemology , ecology , biology
Let F be an algebraically closed field of characteristic p , and let H be a finite group. A natural and important problem in representation theory is to classify the pairs ( G, D ), where G is a subgroup of H , and D is an irreducible FH ‐module that remains irreducible when restricted to G . For example, if H is an almost simple group or a central extension of an almost simple group, then one encounters this problem in the attempt to determine the maximal subgroups of the finite classical groups. We solve the above problem for Schur's double covers H =S ^n andA ^n of the symmetric and alternating groups, and G a Young‐type subgroup. The answer is given in terms of the combinatorics of restricted, p ‐strict partitions; such partitions are used to parameterize the irreducible spin representations ofS ^n andA ^n . We exploit the connections, recently developed by Ariki, Grojnowski, Brundan, Kleshchev, and others, between modular branching rules, crystal graphs of affine Kac–Moody algebras, and affine Hecke algebras and related objects. 2000 Mathematics Subject Classification 20C08, 20C20, 20C25, 20C30 (primary), 17A70, 17B37, 17B67 (secondary)